CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 140



s s

TT == 0

therefore, (42c)

rs , rs ,

B B

Decomposition Eq. (42a) is valuable in the study of electromagnetic an-
gular momentum; here, it came out as a consequence of the spintensor
concept Eq. (40b).
It can be proven that

( )

2

( )

q − 4 b

sar

KwD = = 41.c

, (43a)

4

sar

B , b

where

D

is a tensor employed by Synge [7] in another context:

ijrm

Dg k k = − g k k − g k k − g k k

(43b)

sarb rs a b ab r s ar s b sb a r

Identity Eq. (43a) was obtained by Rowe [53].
Weert did not study Eq. (35a); this analysis was considered in [54-61]
to determine a nonlocal superpotential (it depends on integrals over the
world line) for the radiative part:

a

TK =

(44a)

rs sr a ,

R R

with

)

(

()

τ τ

() () () ()()

∫

σθ

∫

σθβ

K X

i

= qF [ p p a a v d γ + p a a e d γ −

() () ()

sc r r

σθ β scr

0 0

R

(44b)

τ τ

()

σ

∫

2

∫

2

− avd γ − p ae d γ ], σ θ β , , = 1, 2, 3

()

r r

σ

0 0

τ is the proper time in the retarded point

associated to X . Trying out Eq. (44a) brings into relevance the identities
Eq. (24); the integrals in Eq. (44b) indicate the nonlocal character of radia-
tive superpotential; besides, if the four-acceleration

where

( er

is the Fermi triad and

σ

a is annulated, then

= which was to be expected due to Eq. (26). When obtaining Eq. (44),
transport Eq. (9a) is basic; never before had the great value of the Fermi
triad been shown in electrodynamics.

K 0

ijc

R